Codiene is pretty weak.
Hydrocodone is a little higher up there ('vicodin', 'lortab', etc...) but still weak. That stuff would just make me nausous, more than anything.
The most efficient narcotic (pain relief to nausea/respiratory depression ratio), I believe is morphine/heroin.

I was perscribed oxy one time... that works pretty well. Doctors seem to be thrown in jail nowadays if they perscribe it though.
You're prolly being underperscribed. Tylenol w/ codiene seems to be a pretty weak choice for getting teeth pulled. Hell, that stuff is OTC in Canada I believe.

If you wanna ride... I wouldn't take a narcotic if I were you.
Personally I would either ride without pain meds, or ride with Ibprofen.